In view of Proposition REF , the von Neumann algebra
\({M}_{B}\)  turns out to be endowed with the faithful, semi-finite and normal
(FSN) trace \(_B\) . The pair \(({M}_{B},_B)\)  is the basic element for
the development of the noncommutative integration theory in the sense
of [1]}, [2]}, [3]} (see also [4]} and references therein). Let \(T\in \big ({L}^2_B\big )^2\)  be an
element of the domain of \(_B\) . The \(L^p\) -norm of \(T\)  is given by
\({\left|\hspace{-1.0625pt}\left|\hspace{-1.0625pt}\left|T\right|\hspace{-1.0625pt}\right|\hspace{-1.0625pt}\right|}_{B,p} := \left[_B\big (|T|^p\big )\right]^{\frac{1}{p}} ,\qquad 0<p<+\infty .\) 
